You cast spells by wiggling your fingers and saying words. That's how scrolls work. You don't need to remember all these words and actions when they're written down (though, I have no idea why they diappear). Magical talent simply comes from being able to remember what to do.
My speculated guess is that each school requires completely different sets of words. For example, Destruction spells would all have similar words, with slight differences for different spells, whereas Illusion spells would have a completely different set of words, with the same pattern. It's almost like speaking two different languages, hence why each school requires you to remember different words and movements.
Oh, and I don't believe all spells are a simple 5 word chant. The more powerful spells may require quite the essay. Perhaps speaking quicker plays a part, too, cconsidering you cast spells at the same speed? That may just be game mechanics, though.
Well, about the speaking thing... The lore & books suggest it's more to do with a thought process and utilizing magicka to get what you want to happen. I made a big post about the thought processes needed a while ago, based on the "skill books" of various games. For instance, an illusionist woman mentioned that Illusion is really about changing the perception of a target without actually changing it at all, while a person who was skilled in Alteration who was teaching a student told him to basically believe the unbelievable to get it to work well.
Think of it as artistic talent in the real world. Can everyone theoretically draw? Yes. Can everyone draw well? Hell no. You can certainly learn to draw well if you don't have the predisposition, but its going to be harder. Likewise, you don't have to become an artist if you have the talent.
I'm really skeptical when it comes to race having anything to do with magical talent. The Altmer probably breed for it, so they will have more, but the other races don't particularly care either way. There might be a minor improvement in genetic factors based on ones family, but I don't think that looking a specific way makes you more able to use magic. The stats more likely represent the likelihood of someone of the race being a mage (Bretons have more magic users per capita than Redguards, for instance) combined with your characters likelyhood of being good in magic, rather than the intrinsic ability of a race to cast spells.
Art history is full of white dudes. Does that mean that I'm more likely to be an awesome artist because I'm white?
Very good way of putting it! Thank you.
Also, I remember that Bretons have the particular trait of being very good abstract thinkers, which would explain why they'd be particularly able to use stuff like Restoration, Mysticism, Conjuration, and Alteration.